Just send a bill with you name and new address. It's easy, free and is just extra precaution.
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Application self-filed on 8/15 (California); Application sent on 8/15 via FedEx next day; Application arrived 8/16 at 10:30 and signed by EArdnt in Phoenix; G-1145 8/20; I-797C 8/27 (routed from Nebraska); Biometric scheduled 9/12 at 2 PM; Biometric walk-in 8/27 around 2 PM; EAD approval 9/10 ; EAD card arrival 9/14; SSN application 9/14; SSN Card arrival 10/15; DL 10/18